![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > monoordxr | Structured version Visualization version GIF version |
Description: Ordering relation for a monotonic sequence, increasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.) |
Ref | Expression |
---|---|
monoordxr.p | ⊢ Ⅎ𝑘𝜑 |
monoordxr.k | ⊢ Ⅎ𝑘𝐹 |
monoordxr.n | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
monoordxr.x | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ*) |
monoordxr.l | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) |
Ref | Expression |
---|---|
monoordxr | ⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | monoordxr.n | . 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | monoordxr.p | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
3 | nfv 1918 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ (𝑀...𝑁) | |
4 | 2, 3 | nfan 1903 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) |
5 | monoordxr.k | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
6 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
7 | 5, 6 | nffv 6899 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
8 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑘ℝ* | |
9 | 7, 8 | nfel 2918 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) ∈ ℝ* |
10 | 4, 9 | nfim 1900 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ*) |
11 | eleq1w 2817 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ (𝑀...𝑁) ↔ 𝑗 ∈ (𝑀...𝑁))) | |
12 | 11 | anbi2d 630 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ↔ (𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)))) |
13 | fveq2 6889 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
14 | 13 | eleq1d 2819 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℝ* ↔ (𝐹‘𝑗) ∈ ℝ*)) |
15 | 12, 14 | imbi12d 345 | . . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ*) ↔ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ*))) |
16 | monoordxr.x | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ*) | |
17 | 10, 15, 16 | chvarfv 2234 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ*) |
18 | nfv 1918 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ (𝑀...(𝑁 − 1)) | |
19 | 2, 18 | nfan 1903 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (𝑀...(𝑁 − 1))) |
20 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑘 ≤ | |
21 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑘(𝑗 + 1) | |
22 | 5, 21 | nffv 6899 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘(𝑗 + 1)) |
23 | 7, 20, 22 | nfbr 5195 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) ≤ (𝐹‘(𝑗 + 1)) |
24 | 19, 23 | nfim 1900 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑗) ≤ (𝐹‘(𝑗 + 1))) |
25 | eleq1w 2817 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ (𝑀...(𝑁 − 1)) ↔ 𝑗 ∈ (𝑀...(𝑁 − 1)))) | |
26 | 25 | anbi2d 630 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) ↔ (𝜑 ∧ 𝑗 ∈ (𝑀...(𝑁 − 1))))) |
27 | fvoveq1 7429 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑗 + 1))) | |
28 | 13, 27 | breq12d 5161 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)) ↔ (𝐹‘𝑗) ≤ (𝐹‘(𝑗 + 1)))) |
29 | 26, 28 | imbi12d 345 | . . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) ↔ ((𝜑 ∧ 𝑗 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑗) ≤ (𝐹‘(𝑗 + 1))))) |
30 | monoordxr.l | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) | |
31 | 24, 29, 30 | chvarfv 2234 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑗) ≤ (𝐹‘(𝑗 + 1))) |
32 | 1, 17, 31 | monoordxrv 44179 | 1 ⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 Ⅎwnfc 2884 class class class wbr 5148 ‘cfv 6541 (class class class)co 7406 1c1 11108 + caddc 11110 ℝ*cxr 11244 ≤ cle 11246 − cmin 11441 ℤ≥cuz 12819 ...cfz 13481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |